The following question was motivated by my research.
Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs to matrix elements such that the determinant of the matrix will be equal to $1$? I do not know an answer even to a weaker question: is it possible to replace some of the $1$'s in the matrix with odd integers so that the determinant will be equal to $1$?
Remark: it is known that a natural reduction mod $N$ map $SL_n(\mathbb Z) \to SL_n(\mathbb Z/ N\mathbb Z)$ iz is surjective for any $n,N$.
